Simplify the following expression and state the condition under which the simplification is valid. You can assume that $n \neq 0$. $z = \dfrac{27n - 63}{n} \div \dfrac{4(3n - 7)}{n} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $z = \dfrac{27n - 63}{n} \times \dfrac{n}{4(3n - 7)} $ When multiplying fractions, we multiply the numerators and the denominators. $z = \dfrac{ (27n - 63) \times n } { n \times 4(3n - 7) } $ $ z = \dfrac {n \times 9(3n - 7)} {n \times 4(3n - 7)} $ $ z = \dfrac{9n(3n - 7)}{4n(3n - 7)} $ We can cancel the $3n - 7$ so long as $3n - 7 \neq 0$ Therefore $n \neq \dfrac{7}{3}$ $z = \dfrac{9n \cancel{(3n - 7})}{4n \cancel{(3n - 7)}} = \dfrac{9n}{4n} = \dfrac{9}{4} $